Why do people like it? Complexity and depth. Games like DotA (meaning, LoL, HoN, DotA2, etc.) offer a condensed RPG-like experience, with XP, leveling, skills, items and such in a competitive team-based environement. The main draw is the interaction of the wide array of game mechanics - different heroes have drastically different roles and abilities available to them, which in turn means they need to be built differently. Then comes the added dimension of needing to adapt to the flow of the game, your enemies' lineup and builds, your team and your own role in it.
All this results in a more cerebral experience than the average shooter. While player skill plays a major role in the game it is not enough. Knowing the game, the various characters and their abilities, the items, knowing how to recognize the flow of the game and adapt to it (the game is often played radically differently in early, mid and late stages due to how heroes develop over time) - all this is necessary and sometimes more important than pure twitch skill.
If you'd like to learn more about this type of game, I suggest either playing DotA with your friends and having them tutor you, or taking up League of Legends. LoL is probably the best entry-level MOBA game (acronym for DotA style games, means Multiplayer Online Battle Arena) with the gentlest learning curve (all MOBA games have a pretty rough learning period, LoL is just more gentle than others) which can get you familiarized with the genre basics without making you want to murder a kitten. DotA (and DotA2) are on the more unforgiving side of the spectrum and I wouldn't recommend them to new players with no prior experience.